Questions tagged [metric-spaces]
Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.
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The existence of $f$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$ implies the existence of a real $k$-lipschitz function $g$ such that $g|_Y=f$.
Here is the problem from a book for metric spaces I'm trying to solve:
Let $f:Y\rightarrow\mathbb{R}$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$. Prove that there is a $k$-lipschitz function $...
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when do we say a metric space is quasi-invariant under a function?
A measure of a space that is equivalent to itself under "translations" of this space. More precisely: Let $(X,B)$
be a measurable space (that is, a set $X$
with a distinguished $ σ$
-algebra ...
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Preparation for Folland/Royden/Graduate-Level Analysis
I’m trying to prepare for further study in Analysis and was wondering what advice you all would give. I have read (most of) Abbott’s Understanding Analysis and have started Rudin’s Principles of ...
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Hyperbolic metric space and Cayley graph of a group [closed]
The following definition is given in the book "Group Theory from a Geometrical Viewpoint"
Proposition 2.1. The following are equivalent for a geodesic metric space X.
(1) Triangles are slim....
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Problem about fixed points in a complete metric space
Let $(X,d)$ be a non-empty complete metric space and let $ f:X \rightarrow X$ be a function such that for each positive integer $n$ we have
(i) if $ d(x,y)<n+1$ then $d(f(x),f(y))<n$
(ii) if $d(...
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Convex hull of open sets is an open set?
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
...
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Family of sets closed under arbitrary intersections and arbitrary unions of chains of its elements induces a finitary closure operator?
Let $\mathcal{U} \subseteq \wp\left ( X \right )$ be a family of sets that contains both $\emptyset$ and $X$, is closed under arbitrary intersections and is closed under arbitrary unions of chains of ...
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Points in a rectangular configuration define equal metric segments?
Let $(X,d)$ be a metric space. For points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
We ...
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Closure and interior of convex set is convex?
For a metric space $(X,d)$ and points $x,y \in X$ we define the metric segment between them as the following set:
$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$
We then say ...
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Topology basis consisting of convex sets in metric spaces
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
...
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Are convex polytopes closed in arbitrary metric spaces?
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
...
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Prove that the usual metric and other metric induce the same topology
I am working on A course on Borel sets, by S.M. Srivastava. There is this problem I am working on that states the following:
Show that both the metrics $d_1$ and $d_2$ on $\mathbb{R^n}$ defined in 2.1....
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$f,g \in [0,1] , f<g $ , when is $U:=\{h \in C[0,1]:f(t)<h(t)<g(t), \forall t \in [0,1] \}$ a ball in $C[0,1]$ with respect to the sup metric
Let $f,g:[0,1] \to \mathbb R$ be continuous functions such that $f(t)<g(t),\forall t \in [0,1]$ , then under what
additional conditions on $f,g$ can we conclude that
$U:=\{h \in C[0,1]:f(t)<...
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Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.
Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.
How does the following look?
Proof:
For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
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The proof of Birkhoff Kakutani Theorem.
The Theorem: Let $G$ be a topological group. Then $G$ is metrizable iff $G$ is Hausdorff and the identity $1$ has a countable neighborhood basis. Moreover, if $G$ is metrizable, $G$ admits a ...
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